Problem: Determine how many solutions exist for the system of equations. ${-8x-2y = 10}$ ${12x+3y = -15}$
Solution: Convert both equations to slope-intercept form: ${-8x-2y = 10}$ $-8x{+8x} - 2y = 10{+8x}$ $-2y = 10+8x$ $y = -5-4x$ ${y = -4x-5}$ ${12x+3y = -15}$ $12x{-12x} + 3y = -15{-12x}$ $3y = -15-12x$ $y = -5-4x$ ${y = -4x-5}$ Just by looking at both equations in slope-intercept form, what can you determine? ${y = -4x-5}$ ${y = -4x-5}$ Both equations have the same slope and the same y-intercept, which means the lines would completely overlap. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ Since any solution of ${-8x-2y = 10}$ is also a solution of ${12x+3y = -15}$, there are infinitely many solutions.